US9262808B2 - Denoising of images with nonstationary noise - Google Patents
Denoising of images with nonstationary noise Download PDFInfo
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- US9262808B2 US9262808B2 US13/762,022 US201313762022A US9262808B2 US 9262808 B2 US9262808 B2 US 9262808B2 US 201313762022 A US201313762022 A US 201313762022A US 9262808 B2 US9262808 B2 US 9262808B2
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Definitions
- This invention relates generally to image processing, and more particularly to denoising images.
- An image is sparse in the transform domain when most magnitudes of transform domain coefficients are either zero, or negligible. In that case, the image can be well approximated as a linear combination of a small number of bases that correspond to pixel-wise consistent patterns. Denoised image can be obtained by keeping only transform coefficients larger than a first threshold, which are mainly due to the original signal, and discarding coefficients smaller than a second threshold, which are mainly due to noise.
- the sparsity level of an image in the transform domain heavily depends on both the signal and the noise properties.
- the selection of a good sparsity inducing transform is an art, and is effectively a function of the underlying, signal to be denoised, and the noise.
- multi-resolution transforms achieve good sparsity for spatially localized details, such as edges and singularities. Because most images are typically full of such details, transform domain methods have been successfully applied for image denoising.
- Non-local means (NLM) de-noising is based on non-local averaging, of all the pixels in an image.
- the amount of weighting for a pixel is based on a similarity of a small patch of pixels, and another patch of pixels centered on the pixel being dc-noised.
- BM3D block matching in 3D
- PSNR peak signal-to-noise ratio
- block matching in 3D approaches optimal results for constant variance noise, but cannot be improved beyond 0.1 dB values
- BM3D is a two-step process.
- the first step gives an early version of the denoised image by processing stacks of image blocks constructed by block matching.
- the second stage applies a statistical filter in a similar manner.
- For a reference block pixel-wise similar blocks are searched and arranged in a 3D stack.
- an orthogonal transform is applied to the stack, and the noise is reduced by thresholding the transform coefficients, followed by an inverse transform. Sparsity is enhanced due to similarity between the 2D blocks in the 3D stack.
- the second step finds the locations of the blocks similar to the processed block, and forms two groups, one from the noisy image and other from the estimate. Then, the orthogonal transform is applied again to both the groups and Wiener filtering is applied, on the noisy group using an energy spectrum of the estimate as the true energy spectrum.
- the embodiments of the invention provide a method for denoising an image that is corrupted by noise of a spatially varying variance, nonstationary noise.
- the first step is to estimate the noise variance, potentially at every pixel, and then to denoise the image using the estimated variance information.
- the method uses a two-step procedure.
- the first step construct a variance map of the nonstationary noise by solving an optimization problem that is based on a scale invariant property of kurtosis, a measure of the peakedness of the probability distribution of the random noise.
- the second step reconstructs the input image as the output image, patch by patch, using the variance map and collaborative filtering.
- the method performs much better, up to +5 dB, than the state-of-the-art procedures both in the terms of PSNR and a mean structure similarity (MSSIM) index.
- FIG. 1 is a now diagram of a method for denoising an image according to embodiments of the invention.
- FIG. 1 shows a method for denoising an input image 101 that is corrupted by noise of a spatially varying variance, i.e., nonstationary noise.
- a variance map 111 of the nonstationary noise is constructed 110 from the input image by solving an optimization problem with an objective function 105 that is based on a scale invariant property of kurtosis, a measure of the peakedness of the probability distribution of the noise.
- the input image is partitioned 112 into regions 113 that contain overlapping patches.
- the variance map is similarly partitioned such that there is a one-to-one correspondence between the patches.
- a prefilter process 114 is applied to construct an intermediate image 115 . Then, the input image 101 is reconstructed 120 as the output image 121 , patch by patch, using the variance map 111 , the intermediate image 115 and collaborative filtering to produce the denoised output image.
- the method can be performed in a processor 100 connected to memory and input output interlaces as known in the art. It should be noted that our method is autonomous because the only input is the noisy image.
- MIND Multiple Image-Noise Denoising
- a Gaussian variable has a zero kurtosis.
- the noisy input image I n is first transformed to frequency domain.
- a band-pass filtered domain of K channels i.e., the response of the image convolved with K different band-pass filters.
- the kurtosis of an original (noiseless) image and the noisy image in the k th channel are ⁇ k and ⁇ k , respectively.
- ⁇ _ k ⁇ k ( ⁇ _ k 2 - ⁇ 2 ⁇ _ k 2 ) 2 . ( 3 )
- ⁇ 2 ⁇ 2 - ⁇ 1 2
- ⁇ ⁇ ⁇ ⁇ 4 - 4 ⁇ ⁇ 3 ⁇ ⁇ 1 + 6 ⁇ ⁇ 2 ⁇ ⁇ 1 2 - 3 ⁇ ⁇ 1 4 ⁇ 2 2 - 2 ⁇ ⁇ 2 ⁇ ⁇ 1 2 + ⁇ 1 4 . ( 5 )
- a direct approach would estimate the variance and kurtosis for each band of each overlapping image patch of size D ⁇ D using equation (5), where raw moments are estimated using spatial averaging, and then apply the closed form solution of equation (4) to estimate the local noise variance.
- the direct approach is computationally complex. Therefore, we convert the image to an integral image, which makes the moment estimation task a matter of a small number of additions and subtractions.
- the next step denoises the noisy input image.
- the patches are sufficiently small to model noise with a single Gaussian distribution, e.g., 12 ⁇ 12 to 32 ⁇ 32.
- the single noise variance ⁇ p 2 of the p th patch is a weighted mean of the estimated noise variance at every pixel of that patch.
- the noise variance is a maximum of all pixels.
- ncc ⁇ ( I p n , I q n ) [ ⁇ ⁇ ( f 2 ⁇ D ⁇ ( I p n - ⁇ p ) ) ] ⁇ [ ⁇ ⁇ ( f 2 ⁇ D ⁇ ( I q n - ⁇ q ) ) ] ⁇ p ⁇ ⁇ q , where ⁇ is a hard-thresholding operator with a threshold of ⁇ 2D ⁇ p , and f 2D f 2D is DCT. Scaling is done with the spatial domain variance because we are interested in relative scores.
- a prefiltered, intermediate image I m is obtained by mapping back I p (S p ⁇ ) onto the image coordinates and combining the pixel-wise responses, i.e., on a pixel-by-pixel basis, using the weighted mean, where the weights are defined by the local variances
- ⁇ p ⁇ ( ⁇ p 2 ⁇ N ⁇ ⁇ ( p ) ) - 1 if ⁇ ⁇ N ⁇ ⁇ ( p ) ⁇ 1 1 otherwise , ( 7 ) where N ⁇ (p) is the number of the coefficients retained after the hard-thresholding.
- the Wiener deconvolution coefficients in the discrete Fourier transform (DFT) domain are defined from the energy of the transform domain coefficients as
- W ⁇ ( S p w ) ⁇ f 3 ⁇ D ⁇ ( I ⁇ ( S p w ) ) ⁇ 2 ⁇ f 3 ⁇ D ⁇ ( I ⁇ ( S p w ) ) ⁇ 2 + ⁇ p 2 , ( 8 )
- f 3D is the DFT.
- a sparse coding by dictionary learning is used instead of the Wiener filtering for collaborative filtering.
- I p 3D data cluster
- an under-complete dictionary is learned from using an alternative decision process applied to the affinity net.
- the patches in the same cluster are coded by a sparse combination of corresponding dictionary atoms.
- the reconstructed patches are collaboratively aggregated to construct a denoised image, see U.S. application Ser. No. 13/330,795 filed by Assignee.
- Our MIND can be applied to multiplicative noise that is common in radar and laser imaging by operating in a log-intensity domain to transform the multiplicative denoising into additive denoising.
- clusters can be any size, and can be represented by corresponding unique dictionaries that are designed to best represent the coherent variations at the same pixel locations in the cluster data.
- the method takes advantage of kurtosis based local variance estimation and collaborative filtering. It should be noted that the method does not require training, with only input being the noisy image.
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Abstract
Description
I n(i,j)=I(i,j)+η(i,j), (1)
where I(i,j) is the intensity of an image pixel p at location (i,j), and η(i,j) is the noise with a variance σ2(i,j).
κ=
where the variance is σ2=Ex[(x−Ex[x])2], the uncentered 4th order moment is
where the minimizing (min) provides the solution for the variance of the noise. The minimization of equation (4) is possible due its convexivity, and the optimal solution has a closed form.
where φ is a hard-thresholding operator with a threshold of λ2Dσp, and f2D f2D is DCT. Scaling is done with the spatial domain variance because we are interested in relative scores. The result of this step produces a set Sp φ, which contains the coordinates of the patches that are similar to Ip n. We arrange these patches into a 3D structure Ip(Sp φ) on which a 1D transform and hard-thresholding is applied a second time along the patch index, to the values of the pixels at the same patch locations, followed by the inverse 1D transform
Î p(S p φ)=f 1D −1(φ(f 1D(I p(S p φ)))), (6)
where φ is the hard-thresholding operator with a threshold λ1Dσp. The intuition behind this second transform domain hard-thresholding along each pixel is to incorporate support from multiple patches to suppress intensity divergences.
where Nφ(p) is the number of the coefficients retained after the hard-thresholding.
where f3D is the DFT. Here, we also use the previously determined local variances. The element-by-element multiplication in equation (8) with the trans form domain coefficients f3D(Ip(Sp w)) produces the Wiener filtered response in the transform domain, which is then mapped back to the spatial domain by
I p(S p w)=f 3D −1(W(S p w)f 3D(I(S p w))) (9)
to obtain the filtered, patches Ip(Sp w). Then, we project the filtered patches to the output image If to aggregate the multiple estimates for each pixel location with weights inversely proportional to the Wiener coefficients and variance values
ωp w(p)=(σp 2 ∥W(S p w)∥2 2)−1, (10)
so pixels with a larger uncertainty contribute less.
Claims (19)
κ=
I n(i,j)=I(i,j)+η(i,j),
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JP2013250807A JP6057881B2 (en) | 2013-02-07 | 2013-12-04 | Method for removing noise from input image consisting of pixels containing noise |
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US11257189B2 (en) | 2019-05-02 | 2022-02-22 | Samsung Electronics Co., Ltd. | Electronic apparatus and image processing method thereof |
US11861809B2 (en) | 2019-05-02 | 2024-01-02 | Samsung Electronics Co., Ltd. | Electronic apparatus and image processing method thereof |
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US20140219552A1 (en) | 2014-08-07 |
JP6057881B2 (en) | 2017-01-11 |
JP2014154141A (en) | 2014-08-25 |
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